Key understandings in mathematics learning – Paper 5: Understanding space and its representation in mathematics

• Bryant, P. (2007) Key understandings in mathematics learning, Paper 5, Nuffield Foundation, University of Oxford

This paper is about children’s informal knowledge of space and spatial relations as well as their formal learning of geometry. It deals with the connection between these two kinds of knowledge.

Going round in circles

• Clausen-May, T. (2005) Teaching Maths to Pupils with Different Learning Styles, London: Paul Chapman PDF

This article describes how a pair of PowerPoint presentations can be used to establish mental images that can help pupils to understand, and so recall, the principles that underlie the formulae to calculate the circumference and the area of a circle

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Activity A - Thinking 3D

A series of activities designed to support the development of transferring a 2D representation of a 3D object into a model of the object itself as well as gaining an understanding of positional language.

Activity B - Round a Hexagon

An activity from Nrich supporting the finding of missing angles, including internal and external angles and the use of a protractor.

Activity C - Watching the wheels go round and round

An Nrich activity to support understanding of the term circumference. Links are made to converting units of measure.

Activity D - Quadrilaterals Game

An activity to compare and classify geometric shapes based on their properties in a game that is linked to the card game rummy.

Activity E - Property Chart

This activity used property charts to compare and classify geometric shapes based on their properties.

Activity F - Cut Nets

An Nrich activity to support the skills of visualisation in building simple 3-D shapes from nets.

Activity G - Dynamic Geometry

An introduction to dynamic geometry with helpful links to websites offering free software to support the exploration of properties of circles and the formation of nets for a comprehensive selection of 3-D shapes.

Useful Resources

Protractors or angle measurers, rulers, dot isometric paper, pair of compasses, set squares, dynamic geometry software, set of folding geometric shapes, pinboards and geoboards, Perspex mirrors, 2-D shape dice, class sets of ‘polydron’ or other 3D shape construction materials.

Examples of what children should be able to do, in relation to each (boxed) Programme of Study statement

draw 2-D shapes using given dimensions and angles

Children should be able to construct a triangle given two sides and the included angle

Here is a sketch of a triangle. (It is not drawn to scale).

Draw the full size triangle accurately, below. Use an angle measurer (protractor) and a ruler. One line has been drawn for you.

recognise, describe and build simple 3-D shapes, including making nets

Children should be able to identify, visualise and describe properties of rectangles, triangles, regular polygons and 3-D solids; use knowledge of properties to draw 2-D shapes and identify and draw nets of 3-D shapes

They should be able to respond accurately to questions such as;

‘I am thinking of a 3D shape. It has a square base. It has four other faces which are triangles. What is the name of the 3D shape?’

‘Which of these nets are of square based pyramids? How do you know?

‘Is this a net for an open cube?’ How do you know?

compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons

Children should be able to make and draw shapes with increasing accuracy and knowledge of their properties.

They should be able to carry out activities such as;

‘Give me instructions to get me to draw a rhombus using my ruler and a protractor’

‘On the grid below, use a ruler to draw a pentagon that has three right angles’

illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius

They should know that:

• The circumference is the distance round the circle
• The radius is the distance from the centre to the circumference
• The diameter is 2 x radius

recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles

Children should be able to estimate angles, use a protractor to measure and draw them, on their own and in shapes. They should know that the angle sum of a triangle is 180˚, and the sum of angles around a point is 360˚.

They should be able to use this knowledge to respond accurately to questions such as;

‘There are nine equal angles around a point. What is the size of each angle?’

‘There are a number of equal angles around a point. The size of each angle is 24°. How many equal angles are there?’

Children should be able to calculate the size of angle ‘y’ in this diagram without using a protractor.

• KS2 Maths Geometry

This resource from Teachers TV (sign-in required to view) features Richard Dunne, Mathematics consultant, who explains that mathematics involves abstraction and symbolism, the ingredients of higher order mental processes, an essential skill that students need to develop.

• Nrich Air Nets

This series of 24 very short video clips (each lasting a few seconds) challenges children to determine whether a given net can be assembled to make a 3D shape. A good, interactive resources for use in the classroom.